Preprint Stephan Dempe and Patrick Mehlitz Lipschitz continuity of the optimal value function in parametric optimization ISSN

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1 Fakultät für Mathematik und Informatik Preprint Stephan Dempe and Patrick Mehlitz Lipschitz continuity of the optimal value function in parametric optimization ISSN

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3 Stephan Dempe and Patrick Mehlitz Lipschitz continuity of the optimal value function in parametric optimization TU Bergakademie Freiberg Fakultät für Mathematik und Informatik Prüferstraße FREIBERG

4 ISSN Herausgeber: Herstellung: Dekan der Fakultät für Mathematik und Informatik Medienzentrum der TU Bergakademie Freiberg

5 Lipschitz continuity of the optimal value function in parametric optimization S. Dempe, P. Mehlitz Faculty of Mathematics and Computer Science Technical University Bergakademie Freiberg Abstract We study generalized parametric optimization problems in Banach spaces, given by continuously Fréchet differentiable mappings and some abstract constraints, in terms of local Lipschitz continuity of the optimal value function. Therefore we make use of the well-known regularity condition by Kurcyusz, Robinson and Zowe, an inner semicontinuity property of the solution set mapping and some earlier results by Mordukhovich, Nam and Yen. The main theorem presents a handy formula which can be used in order to approximate the Clarke subdifferential of the optimal value function, provided that the conditions mentioned above are satisfied and hence the optimal value function is locally Lipschitz continuous. Throughout the paper we avoid any compactness assumptions. Keywords: Parametric optimization, variational analysis, Banach space, optimal value function, Lipschitz continuity, inner semicontinuity, Clarke subdifferential 1 Introduction and notations In order to study stability properties of parametric optimal control problems it is necessary to analyse general parametric optimization problems such as inf y {f(x, y) g(x, y) K, h(x, y) = o Z, y Ω} (1) where f : X Y R, g : X Y W and h: X Y Z are continuously Fréchet differentiable mappings between real Banach spaces W, X, Y and Z. Furthermore, o Z denotes the zero vector of Z and K W is a non-empty, closed, convex and pointed cone while Ω Y is a non-empty, closed and convex set. We are interested in deriving continuity properties of the optimal value function ϕ: X R and the solution set mapping Ψ: X P(Y ) of (1) given respectively by inf{f(x, y) y Γ(x)} if Γ(x) x X : ϕ(x) = y (2) + if Γ(x) = and x X : Ψ(x) = {y Γ(x) f(x, y) = ϕ(x)} (3) where P(Y ) denotes the power-set of Y and Γ: X P(Y ) is the feasible set mapping of (1) defined as follows: x X : Γ(x) = {y Y g(x, y) K, h(x, y) = o Z, y Ω}. Note that for some x X the set Ψ(x) might be empty although Γ(x) is non-empty and ϕ(x) is a finite number. This problem may arise since we used the infimum to define the function ϕ in (2) and claim in (3) that this infimum is obtained. Some general results on the continuity properties of ϕ and Ψ as well as an introduction to the 1

6 theory of continuity properties of set-valued mappings can be found in [2]. Problem (1) has been studied in view of the directional differentiability of ϕ for example by Shapiro in [14] and by Bonnans and Shapiro in [3] likewise, using some compactness assumptions that are difficult to guarantee if the Banach spaces mentioned above are infinite-dimensional. In this paper we are interested in finding conditions that ensure the Lipschitz continuity of ϕ around a fixed point x X. This problem has already been dealt with by Dempe [5], considering the case where Γ has a compact graph. More research on this topic has been done by Mordukhovich [10] and Mordukhovich et al. [11], using geometric constraint qualifications and the coderivative-principle to derive several very general results. The question arises whether these results can be adapted in order to fit problem (1). We will answer this question, exploiting the well-known regularity condition of Kurcyusz and Zowe [8] as well as Robinson [13], and an inner semicontinuity property of Ψ at a point of its graph. This way we can avoid any kind of compactness assumption which means that our result can be applied even in the case where the Banach spaces W, X and Y are infinite-dimensional. Now we are going to present several concepts and notations used throughout this paper. First of all let N, R, R + and R + 0 denote the natural numbers without zero, the real numbers, the positive real numbers and the non-negative real numbers, respectively. For a real Banach space X with norm X and a non-empty set A X the sets int(a), cl(a), aff(a), L(A), conv(a) and cone(a) denote the interior, the closure, the affine hull, the linear hull, the convex hull and the conic hull of A, respectively. Additionally, X expresses the dual space of X and, : X X R the corresponding dual pairing. Let (x k ) and (p k ) be any sequences of X and X, respectively. Then x k x denotes the convergence of (x k ) to x X while p k p denotes the weak convergence of (p k ) to p X. Let B X be the closed unit ball and U X the open unit ball of X. Furthermore, let U ε X ( x) := ε U X + { x} for any x X and ε R +. The space X is Asplund if the dual space U of every separable subspace U of X is separable. Note that every reflexive Banach space is an Asplund space (cf. [12]). Let Y be another Banach space. The set L(X, Y ) contains all bounded, linear operators from X to Y and is a Banach space as well (cf. [15]). Let F be an element of L(X, Y ). Then F denotes its adjoint and F [x] the image of x X under F. Moreover, ker(f ) = {x X F [x] = o Y } is the kernel of F. Let A D := {p X x A: x, p 0} be the dual cone of the set A. The following definitions and facts are taken from chapter 1 of [10]. For any x X and ε R + 0 we define the set N A ( x, ε) of all ε-normals to A at x as follows: N A ( x, ε) := { p X lim sup x x, x A } x x, p ε. x x X It is always a closed and convex set depending on the chosen norm X. Furthermore, the basic normal cone (often also referred to as Mordukhovich normal cone) NA B ( x) and the normal cone N A ( x) to A at x are defined respectively as: N B A ( x) := {p X (x k ) A (ε k ) R + 0 (p k) X : N A ( x) := {p X x A: x x, p 0}. x k x, ε k 0, p k p, p k N A (x k, ε k ) k N} It is worth to mention that in the definition of the basic normal cone the limiting procedure ε k 0 can be dropped, if X is an Asplund space. That means in this case we have: N B A ( x) = {p X (x k ) A (p k ) X : x k x, p k p, p k N A ( x, 0) k N}. 2

7 Obviously, N A ( x) = (A { x}) D always holds. Let A be convex. It follows N A ( x, ε) = {p X x X : x x, p ε x x X } as well as N A ( x, 0) = NA B( x) = N A( x). Besides, let N A ( x, ε) = NA B( x) = N A( x) = for all x / A. Throughout this paper we will refer to the Fréchet derivative of g at a point ( x, ŷ) as g ( x, ŷ). Additionally, g x( x, ŷ) and g y( x, ŷ) denote the partial Fréchet derivatives of g at ( x, ŷ) with respect to x and y. Let ψ : X R be a lower semicontinuous functional. We call the set B ψ( x) := {p X } (p, 1) Nepi(ψ) B ( x, ψ( x)), where epi(ψ) = {(x, α) X R ψ(x) α} denotes the epigraph of ψ, basic subdifferential of ψ at x (cf. [10]). Note that epi(ψ) is closed because ψ is lower semicontinuous. Besides, if ψ is a locally Lipschitz continuous functional, we define its upper Clarke directional derivative ψ ( x; d) at x in direction d X according to: Moreover, the set ψ ( x; d) := lim sup x x, t 0 ψ(x + td) ψ(x). t c ψ( x) := {p X d X : d, p ψ ( x; d)} is called Clarke subdifferential of ψ at x. The set c ψ( x) is non-empty, bounded, closed and convex (cf. [4]). If X is considered to be an Asplund space, it follows from Theorem 3.57 in [10] that cl ( conv ( B ψ( x) )) = c ψ( x) holds, where cl represents the weak closure in X. If ψ is convex, the sets c ψ( x) and B ψ( x) both coincide with the subdifferential of convex analysis. Finally, we have B ψ( x) = c ψ( x) = {ψ ( x)} provided ψ is continuously Fréchet differentiable at x. Let Θ: X P(Y ) be a set-valued mapping. The sets dom(θ) = {x X Θ(x) } and graph(θ) = {(x, y) X Y y Θ(x)} are called domain and graph of Θ, respectively. Furthermore, we say that Θ is non-trivial if its domain is non-empty. The mapping Θ is said to be inner semicontinuous at a point ( x, ŷ) graph(θ) (cf. [6]) if for any sequence (x k ) X fulfilling x k x there exists an index k 0 N and a sequence (y k ) Y so that y k ŷ and y k Θ(x k ) holds for all k k 0. It follows from the definition that Θ is inner semicontinuous at all points ( x, y) graph(θ) if Θ is lower semicontinuous in the sense of Berge at x (cf. [2]). Moreover, Θ is called locally Lipschitz-like at ( x, ŷ) (cf. [10]) if there exist neighbourhoods U of x and V of ŷ as well as a constant L R + so that x 1, x 2 U : Θ(x 1 ) V Θ(x 2 ) + L x 1 x 2 X B Y holds. One may easily check that Θ is inner semicontinuous at ( x, ŷ) if it is locally Lipschitz-like at this point. More interesting facts on the topic of set-valued mappings can be found in [1]. All of the following concepts and results are taken from Mordukhovich again (cf. [10]). We call the set-valued mapping DN Θ( x, ŷ): Y P(X ), which is defined as q Y : DNΘ( x, ŷ)(q) := {p X } (p, q) Ngraph(Θ) B ( x, ŷ), 3

8 normal coderivative of the set-valued mapping Θ at ( x, ŷ). As above, let DN Θ( x, ỹ)(q) = for any q Y if ( x, ỹ) is not an element of graph(θ). Finally, we call the set A X sequentially normally compact or SNC at a point x A if for any sequences (x k ) A, (ε k ) R + 0 and (p k ) X, fulfilling x k x, ε k 0, p k o X and p k N A (x k, ε k ) for all k N, especially p k o X holds. If X is finite-dimensional, the set A is SNC at all of its points. An important result by Mordukhovich (Theorem 1.21 in [10]) revealed that every convex set A with non-empty relative interior is SNC at any point x A if and only if the codimension of A is finite. That means, the dimension of the factor space X/ cl(aff(a) {x}) is finite for some x A. As a consequence, a set with just one element x is SNC at this point if and only if X is finite dimensional. Let u: X Y and v : Y Z be mappings between Banach spaces X, Y and Z. Then u v : X Z denotes the composition of u and v defined by (u v)(x) = v(u(x)) for any x X. The following lemma is based on some basic results from functional analysis and will be handy in order to construct an approximation formula for the Clarke subdifferential of the optimal value function. Since it has no background in optimization theory we just mention it here. Lemma 1.1 Let E and F be Banach spaces and assume that p E and q L(E, F ) are bounded linear operators. Furthermore, let A F be a non-empty and convex set. Consider the set Q = {p + q a E a A}. Then the following holds true: (i) The set Q is non-empty and convex. (ii) If F is reflexive and A is additionally bounded and closed, then Q is weakly compact. Proof: (i) Obviously, Q is a non-empty set since A is non-empty. Let us take arbitrary elements b 1, b 2 Q and some scalar α [0, 1]. Then there are a 1, a 2 A such that b i = p + q a i for i = 1, 2. That is why we get: α b 1 + (1 α) b 2 = α (p + q a 1 ) + (1 α) (p + q a 2 ) = p + q (α a 1 + (1 α) a 2 ). Since A is convex we deduce α a 1 + (1 α) a 2 A and hence α b 1 + (1 α) b 2 Q. As a result, Q is a convex set. (ii) Let (b k ) Q be an arbitrary sequence. Then there must be a sequence (a k ) A such that b k = p + q a k holds true for all k N. Since A is a bounded, closed and convex subset of the reflexive Banach space F it is weakly compact (cf. [7]). That is why there exists a subsequence (a kn ) (a k ) converging weakly to some a A. As a consequence we get for any x E x, q a kn = q(x), a kn kn q(x), a = x, q a since (a kn ) especially converges weakly to a. We conclude that (q a kn ) converges weakly to q a which additionally implies that (b kn ) converges weakly to p + q a Q. Since (b k ) contains a weakly convergent subsequence whose weak limit belongs to Q and (b k ) was chosen arbitrarily, Q is weakly compact. # 4

9 2 KRZCQ and PKRZCQ The most important regularity condition used in infinite-dimensional optimization is undoubtedly the one by Kurcyusz, Robinson and Zowe (cf. [8], [13]). Throughout this section let G: X Y W Z be the mapping defined as follows: (x, y) X Y : G(x, y) = (g(x, y), h(x, y)). Definition 2.1 (KRZCQ) (cf. [7]) Considering (1) it is said that KRZCQ, the Kurcyusz- Robinson-Zowe-Constraint-Qualification, holds at a feasible point ( x, ŷ) graph(γ) if the following condition is satisfied: (o W, o Z ) int ( {G( x, ŷ)} + G ( x, ŷ)[x Ω {( x, ŷ)}] + K {o Z } ). (4) Let KRZCQ hold at a feasible point ( x, ŷ) graph(γ). Then (4) easily leads to: (o W, o Z ) int ( G ( x, ŷ)[cone(x Ω {( x, ŷ)})] + cone(k {o Z } + {G( x, ŷ)}) ). Due to the fact that the set on the right-hand side is the interior of a cone we get: W Z = G ( x, ŷ)[cone(x Ω {( x, ŷ)})] + cone(k {o Z } + {G( x, ŷ)}). (5) Applying the open-mapping-theorem (cf. Theorem IV.3.3 in [15]) and Theorem 2.1 in [8] provides that (5) implies (4). That is why KRZCQ holds at ( x, ŷ) if and only if (5) is satisfied. The following lemma gives another characterization of KRZCQ, which is especially useful for proofs. Lemma 2.1 Let ( x, ŷ) graph(γ) be an arbitrary feasible point of (1). Then KRZCQ holds at ( x, ŷ) if and only if the following conditions are satisfied: W = g ( x, ŷ)[cone(x Ω {( x, ŷ)}) ker(h ( x, ŷ))] + cone(k + {g( x, ŷ)}) Z = h ( x, ŷ)[cone(x Ω {( x, ŷ)})]. (6) Proof: Due to the argumentation above we just have to show the equivalence of (5) and (6). [= ] Let (5) hold. Obviously, we get h ( x, ŷ)[cone(x Ω {( x, ŷ)})] = Z. Moreover, for any (w, o Z ) W Z there exists d cone(x Ω {( x, ŷ)}) and ϑ cone(k + {g( x, ŷ)}) so that w = g ( x, ŷ)[d] + ϑ and o Z = h ( x, ŷ)[d] are satisfied. That is why d is an element of cone(x Ω {( x, ŷ)}) ker(h ( x, ŷ)) and finally (6) holds true. [ =] Let the conditions in (6) be satisfied and (w, z) W Z be chosen arbitrarily. Due to (6) there is a d cone(x Ω {( x, ŷ)}) so that z = h ( x, ŷ)[d] holds. Let w = g ( x, ŷ)[d]. Again we use (6) to derive the existence of d cone(x Ω {( x, ŷ)}) ker(h ( x, ŷ)) and ϑ cone(k + {g( x, ŷ)}) so that w w = g ( x, ŷ)[d ] + ϑ is satisfied. Finally, let d X Y and ϑ W Z be given by d = d + d cone(x Ω {( x, ŷ)}) and ϑ = (ϑ, o Z ) cone(k {o Z } + {G( x, ŷ)}). Then we can deduce G ( x, ŷ)[ d] + ϑ = (g ( x, ŷ)[d] + g ( x, ŷ)[d ] + ϑ, h ( x, ŷ)[d] + h ( x, ŷ)[d ]) = (w + w w, z) = (w, z) since d is an element of ker(h ( x, ŷ)). As a consequence (5) must hold. # Another important lemma which will be used in order to approximate the normal coderivative of the mapping Γ at a point of its graph will be presented now. 5

10 Lemma 2.2 Let ( x, ŷ) graph(γ) be a feasible point of (1) where KRZCQ holds. following regularity conditions are satisfied: Then the (K {o Z } + {G( x, ŷ)}) D ker(g ( x, ŷ) ) = {o (W Z) }, (7) G ( x, ŷ) [ (K {o Z } + {G( x, ŷ)}) D] ( N X Ω ( x, ŷ)) = {o (X Y ) }. (8) Proof: To show (7) we take an arbitrary element l of (K {o Z } + {G( x, ŷ)}) D ker(g ( x, ŷ) ). Since l (K {o Z } + {G( x, ŷ)}) D holds, for any ξ K and α R + 0 we get α((ξ, o Z) + G( x, ŷ)), l = α (ξ, o Z ) + G( x, ŷ), l 0. Furthermore, l ker(g ( x, ŷ) ) leads to G ( x, ŷ) [l] = o (X Y ) which is, by definition of the adjoint, equivalent to G ( x, ŷ) l = o (X Y ). Putting both observations together with KRZCQ and (5) provides: l[w Z] = l [ G ( x, ŷ)[cone(x Ω {( x, ŷ)})] + cone(k {o Z } + {G( x, ŷ)}) ] = { ζ, G ( x, ŷ) l + α((ξ, o Z ) + G( x, ŷ)), l ζ cone(x Ω {( x, ŷ)}), α R + 0, ξ K} = { α((ξ, o Z ) + G( x, ŷ)), l α R + 0, ξ K} R + 0 This implies l[w Z] = {0} since l[w Z] is a subspace of R. That is why l equals o (W Z) and (7) holds. In order to verify (8) we take some k G ( x, ŷ) [ (K {o Z } + {G( x, ŷ)}) D] ( N X Ω ( x, ŷ)). Then there is l (K {o Z }+{G( x, ŷ)}) D so that k = G ( x, ŷ) [l] and, by definition of the adjoint, k = G ( x, ŷ) l hold. Furthermore, we derive α((x, y) ( x, ŷ)), k = α (x, y) ( x, ŷ), k 0 for all α R + 0 and all (x, y) X Ω since k is an element of N X Ω( x, ŷ). Taking into account that l (K {o Z } + {G( x, ŷ)}) D holds, for any ϑ cone(k {o Z } + {G( x, ŷ)}) and any ζ cone(x Ω {( x, ŷ)}) this leads to: 0 ζ, k = ζ, G ( x, ŷ) l = G ( x, ŷ)[ζ], l G ( x, ŷ)[ζ], l + ϑ, l = G ( x, ŷ)[ζ] + ϑ, l. Besides, we get from KRZCQ { G ( x, ŷ)[ζ] + ϑ } = W Z, ϑ cone(k {o Z }+{G( x,ŷ)}) ζ cone(x Ω {( x,ŷ)}) so that the inequality above shows l[w Z] R + 0. Since l[w Z] is a subspace of R we derive l = o (W Z). Finally, we use the linearity of G ( x, ŷ) to come up with k = G ( x, ŷ) [l] = o (X Y ). That is why (8) holds. # In order to formulate necessary optimality conditions for (1), respecting the role of x as a parameter, it is possible to discuss the following Fritz-John-conditions (Theorem 3.1 and the following remarks in [9]): Let ( x, ŷ) be a local optimal solution of (1) (that means the parameter x X is fixed and ŷ is a local optimal solution of the non-parametric problem (1)). Furthermore, let the so called Fritz- John-conditions be satisfied at ( x, ŷ). That means there exist λ 0 R + 0, λ KD and µ Z, which are not all simultaneously zero, so that the following conditions hold: d cl(cone(ω {ŷ})): (λ 0 f y( x, ŷ) + g y( x, ŷ) λ + h y( x, ŷ) µ)[d] 0, (9) g( x, ŷ), λ = 0. (10) 6

11 The triple (λ 0, λ, µ) is called Lagrange-multiplier of (1) at ( x, ŷ). In order to ensure not only the existence of such a Lagrange-multiplier but also the positivity of λ 0 it is necessary to claim a constraint qualification to hold at ( x, ŷ). Due to the fact that KRZCQ does not respect the role of x as a parameter it is insufficient for that purpose. Instead we use a slightly stronger version of this regularity condition. Definition 2.2 (PKRZCQ) (cf. [3]) Considering (1) it is said that PKRZCQ, the Parametric- Kurcyusz-Robinson-Zowe-Constraint-Qualification, holds at a feasible point ( x, ŷ) graph(γ) if the following condition is satisfied: (o W, o Z ) int ( {G( x, ŷ)} + G y( x, ŷ)[ω {ŷ}] + K {o Z } ). (11) Note that similar to the proof of Lemma 2.1 it can be shown that PKRZCQ holds at a feasible point ( x, ŷ) graph(γ) if and only if the following conditions are satisfied: W = g y( x, ŷ)[cone(ω {ŷ}) ker(h y( x, ŷ))] + cone(k + {g( x, ŷ)}) Z = h y( x, ŷ)[cone(ω {ŷ})]. (12) Using Theorem 3.2 of [9] it is easy to see, that λ 0 = 1 can actually be chosen in (9), provided PKRZCQ holds at ( x, ŷ). The set Λ( x, ŷ) = {(λ, µ) K D Z (1, λ, µ) is Lagrange-multiplier} is called set of all regular Lagrange-multipliers of (1) at ( x, ŷ). It is non-empty, bounded, closed and convex, provided ( x, ŷ) is a local optimal solution of (1) where PKRZCQ holds (cf. [8]). As mentioned above, PKRZCQ is a stronger regularity condition than KRZCQ. Indeed, if PKRZCQ holds at a feasible point ( x, ŷ) graph(γ) we derive with (12) g ( x, ŷ)[cone(x Ω {( x, ŷ)}) ker(h ( x, ŷ))] + cone(k + {g( x, ŷ)}) as well as g ( x, ŷ) [ (X cone(ω {ŷ})) ( ker(h x( x, ŷ)) ker(h y( x, ŷ)) )] + cone(k + {g( x, ŷ)}) = g ( x, ŷ)[ker(h x( x, ŷ)) (cone(ω {ŷ}) ker(h y( x, ŷ)))] + cone(k + {g( x, ŷ)}) = g x( x, ŷ)[ker(h x( x, ŷ))] + g y( x, ŷ)[cone(ω {ŷ}) ker(h y( x, ŷ))] + cone(k + {g( x, ŷ)}) = g x( x, ŷ)[ker(h x( x, ŷ))] + W = W h ( x, ŷ)[cone(x Ω {( x, ŷ)})] = h ( x, ŷ)[x cone(ω {ŷ})] = h x( x, ŷ)[x] + h y( x, ŷ)[cone(ω {ŷ})] = h x( x, ŷ)[x] + Z = Z. Now it follows from Lemma 2.1 that KRZCQ holds at ( x, ŷ). The next lemma will be used in order to formulate a criterion which ensures the property of Γ to be locally Lipschitz-like at a point of its graph. Lemma 2.3 Let ( x, ŷ) graph(γ) be a feasible point of (1) where PKRZCQ holds. Then the following regularity condition is satisfied: p X : (p, o Y ) G ( x, ŷ) [ (K {o Z } + {G( x, ŷ)}) D] + N X Ω ( x, ŷ) = p = o X (13) Proof: First we want to point out that the spaces (X Y ) and X Y are isomorphic which allows us to identify any element of (X Y ) with a pair in X Y. That is why we can choose some element (p, o Y ) G ( x, ŷ) [ (K {o Z } + {G( x, ŷ)}) D] +N X Ω ( x, ŷ). In other words there exist 7

12 l (K {o Z } + {G( x, ŷ)}) D and q N Ω (ŷ) so that (p, o Y ) = G ( x, ŷ) [l] + (o X, q) holds. From the definition of the adjoint we get for any (x, y) X Y : That is why (x, y), (p, o Y ) = (x, y), G ( x, ŷ) [l] + (o X, q) = G ( x, ŷ)[x, y], l + y, q = G x( x, ŷ)[x], l + G y( x, ŷ)[y], l + y, q. (x, y) X Y : x, p = G x( x, ŷ)[x], l + G y( x, ŷ)[y], l + y, q must hold. This is only possible if y Y : 0 = G y( x, ŷ)[y], l + y, q is satisfied. The next formula follows immediately: y Y ϑ cone(k {o Z } + {G( x, ŷ)}): ϑ, l = G y( x, ŷ)[y] + ϑ, l + y, q. (14) As a consequence of PKRZCQ we have: { G y ( x, ŷ)[y] + ϑ } = W Z. ϑ cone(k {o Z }+{G( x,ŷ)}) y cone(ω {ŷ}) Taking into account that l is an element of (K {o Z }+{G( x, ŷ)}) D and that q belongs to N Ω (ŷ) we reason with (14): { l[w Z] = G y ( x, ŷ)[y] + ϑ, l } = ϑ cone(k {o Z }+{G( x,ŷ)}) y cone(ω {ŷ}) ϑ cone(k {o Z }+{G( x,ŷ)}) y cone(ω {ŷ)} { ϑ, l y, q } R + }{{}}{{} Again, since l[w Z] is a subspace of R we conclude l[w Z] = {0} and l = o (W Z), consequently. That is why (14) gives us q = o Y and due to the linearity of G ( x, ŷ) we derive p = o X. Hence the regularity condition (13) is satisfied. # The following corollary is an immediate consequence of Lemma 2.1 and some results by Mordukhovich (Theorems 4.31 and 4.32 in [10]). Corollary 2.1 Let ( x, ŷ) graph(γ) be a feasible point of (1). (i) If the operator G ( x, ŷ) is surjective and Ω = Y holds, the following equality is satisfied: q Y : D NΓ( x, ŷ)(q) = { p X (p, q) G ( x, ŷ) [ (K {o Z } + {G( x, ŷ)}) D]}. (ii) If the spaces W, X as well as Y are Asplund, K is SNC at g( x, ŷ), Z is finite-dimensional and KRZCQ holds at ( x, ŷ), the following is true: q Y : D NΓ( x, ŷ)(q) { p X (p, q) G ( x, ŷ) [ (K {o Z } + {G( x, ŷ)}) D] + N X Ω ( x, ŷ) }. 8

13 Proof: (i) First observe that due to the convexity of K we get: N B ( K) {o Z } (G( x, ŷ)) = N ( K) {o Z }(G( x, ŷ)) = {(r, s) W Z (w, z) ( K) {o Z }: w g( x, ŷ), r + z h( x, ŷ), s 0} = {r W w K : w g( x, ŷ), r 0} Z = {r W v K : v ( g( x, ŷ)), r 0} Z = ( N K ( g( x, ŷ)) ) Z = (K + {g( x, ŷ)}) D Z = (K {o Z } + {G( x, ŷ)}) D. Since all mappings mentioned in (1) are considered to be continuously Fréchet differentiable, the statement immediately follows from statement (i) of Theorem 4.31 in [10]. (ii) As above we get N B ( K) {o Z } (G( x, ŷ)) = (K {o Z} + {G( x, ŷ)}) D from the convexity of K. Additionally, we can deduce N B X Ω ( x, ŷ) = N X Ω( x, ŷ) from the convexity of Ω. We make use of Lemma 2.2 in order to verify that (7) and (8) are satisfied since KRZCQ holds at ( x, ŷ). Moreover, Z is finite-dimensional and K is SNC at g( x, ŷ) which implies that K {o Z } is SNC at G( x, ŷ). Again, we point out that all mappings mentioned in (1) are continuously Fréchet differentiable so that the statement follows from Theorem 4.32(d) in [10]. # As mentioned above we can use Lemma 2.3 and Theorem 4.37 in [10] to formulate a criterion which is sufficient for the property of Γ to be locally Lipschitz-like at a point of its graph. Corollary 2.2 Let ( x, ŷ) graph(γ) be a feasible point of (1) where PKRZCQ is satisfied. Furthermore, let X and Y be Asplund spaces, let Z be finite-dimensional and let K be SNC at g( x, ŷ). (i) If the operator G ( x, ŷ) is surjective and Ω = Y holds, then Γ is locally Lipschitz-like at ( x, ŷ). (ii) If W is an Asplund space, then Γ is locally Lipschitz-like at ( x, ŷ). Proof: Similar to the proof of Corollary 2.1 it can be shown that N B X Ω ( x, ŷ) = N X Ω( x, ŷ) and N B ( K) {o Z } (G( x, ŷ)) = (K {o Z} + {G( x, ŷ)}) D hold, whereas K {o Z } is SNC at G( x, ŷ). Statement (i) follows from statement (i) of Theorem 4.37 in [10] since g and h are assumed to be continuously Fréchet differentiable. Moreover, due to Lemma 2.3 the regularity condition (13) is satisfied at ( x, ŷ) and additionally we have N X Y ( x, ŷ) = {(o X, o Y )}. In order to show statement (ii) we point out that, since PKRZCQ implies KRZCQ, we can deduce from Lemma 2.2 that (7) and (8) are satisfied at ( x, ŷ). Furthermore, we use Lemma 2.3 to verify that the regularity condition (13) holds at ( x, ŷ). Finally, we apply statement (ii) of Theorem 4.37 in [10] which gives us the desired result since g and h are continuously Fréchet differentiable. # Note that if in (i) the operator G y( x, ŷ) is assumed to be surjective, then the operator G ( x, ŷ) is surjective as well and as a consequence PKRZCQ automatically holds at ( x, ŷ) since Ω = Y is satisfied. 9

14 3 Lipschitz continuity of the optimal value function First of all, we want to give a lemma which ensures the Lipschitz continuity of the optimal value function without considering the special structure of the feasible set mapping Γ. Lemma 3.1 We consider problem (1). Let x int(dom(ψ)) be chosen in such a way, that there exists a point ( x, ŷ) graph(ψ) where Γ is locally Lipschitz-like and Ψ is inner semicontinuous. Then the optimal value function ϕ is Lipschitz continuous around x. Proof: Since f is assumed to be continuously Fréchet differentiable there exists γ R + so that f is Lipschitz continuous with Lipschitz modulus L f R + on the set U γ X Y ( x, ŷ). Due to the fact that Γ is locally Lipschitz-like at ( x, ŷ) there are ε, δ, L Γ R + so that x 1, x 2 U ε X( x): Γ(x 1 ) U δ Y (ŷ) Γ(x 2 ) + L Γ x 1 x 2 X B Y (15) holds. Moreover, there is a constant β R + which fulfills U β X ( x) dom(ψ). We assume w.l.o.g. that ε β and max{ε; 2εL Γ + δ} γ hold. Let x 1, x 2 U ε X ( x) be chosen arbitrarily. Due to the inner semicontinuity of Ψ at ( x, ŷ) we can choose y 1 Ψ(x 1 ) which fulfills y 1 U δ Y (ŷ) if only ε is sufficiently small. Using (15) we can find y 2 Γ(x 2 ) so that y 1 y 2 Y L Γ x 1 x 2 X is satisfied. Observing the choice of ε and δ we derive: (x 1, y 1 ) ( x, ŷ) X Y = max{ x 1 x X ; y 1 ŷ Y } < max{ε; δ} max{ε; 2εL Γ + δ} γ (x 2, y 2 ) ( x, ŷ) X Y = max{ x 2 x X ; y 2 ŷ Y } max{ x 2 x X ; y 2 y 1 Y + y 1 ŷ Y } max{ x 2 x X ; L Γ x 2 x 1 X + y 1 ŷ Y } < max{ε; 2εL Γ + δ} γ. That is why the points (x 1, y 1 ) and (x 2, y 2 ) belong to U γ X Y ( x, ŷ). Since f is Lipschitz continuous on that set, we finally get: ϕ(x 2 ) ϕ(x 1 ) f(x 2, y 2 ) f(x 1, y 1 ) L f max{ x 2 x 1 X ; y 2 y 1 Y } L f max{ x 2 x 1 X ; L Γ x 2 x 1 X } = max{l f ; L f L Γ } x 2 x 1 X. Changing the roles of x 1 and x 2 gives analogously: ϕ(x 1 ) ϕ(x 2 ) max{l f ; L f L Γ } x 2 x 1 X. Hence ϕ is Lipschitz continuous around x with Lipschitz modulus max{l f ; L f L Γ }. # Note that there are no compactness assumptions (like in [3] or [5]) necessary in order to prove Lemma 3.1. It is worth to notice that the condition x int(dom(ψ)) is rather strong but can be satisfied, postulating Γ to have non-empty, bounded, closed and convex images in a neighbourhood U of x, f(x, ) to be weakly lower semicontinuous for all parameters in x U and Y to be a reflexive Banach space (Theorem 2.3 in [7]). Lemma 3.1 points out that the property of Γ to be locally Lipschitz-like at a point of its graph 10

15 is somehow important in order to guarantee the Lipschitz continuity of ϕ around a fixed point. Luckily Corollary 2.2 provides a sufficient condition for Γ to have this property. Besides, we make use of Corollary 2.1 and a theorem by Mordukhovich et al. (cf. [11]) in order to formulate our main result. Theorem 3.1 We consider problem (1). Let x int(dom(ψ)) be chosen in such a way, that there exists a point ( x, ŷ) graph(ψ) where Ψ is inner semicontinuous and PKRZCQ holds. Let the spaces X and Y be Asplund whereas Z is finite-dimensional. Furthermore, let K be SNC at g( x, ŷ). Then ϕ is Lipschitz continuous around x, provided that one of the following conditions is satisfied. (i) The operator (g ( x, ŷ), h ( x, ŷ)) is surjective and Ω = Y holds. (ii) The space W is Asplund. Additionally, in both cases the approximation formula c ϕ( x) cl ({ f x( x, ŷ) + g x( x, ŷ) λ + h x( x, ŷ) µ (λ, µ) Λ( x, ŷ) }) (16) holds and cl can be dropped in (16) if the Banach space W is reflexive. Proof: (i) Since all assumptions of Corollary 2.2 (statement (i)) are satisfied with respect to ( x, ŷ), Γ is locally Lipschitz-like at this point. Using Lemma 3.1 we come up with the local Lipschitz continuity of ϕ at x. That is why we just have to show formula (16). Firstly, we make use of statement (i) of Theorem 7 in [11] which provides B ϕ( x) {f x( x, ŷ)} + D NΓ( x, ŷ)(f y( x, ŷ)) since Ψ is inner semicontinuous at ( x, ŷ). Due to the fact that X is an Asplund space we derive: c ϕ( x) = cl ( conv ( B ϕ( x) )) cl ( conv ( {f x( x, ŷ)} + D NΓ( x, ŷ)(f y( x, ŷ)) )). (17) Let a be an arbitrary element of {f x( x, ŷ)} + DN Γ( x, ŷ)(f y( x, ŷ)). Then we can find â DN Γ( x, ŷ)(f y( x, ŷ)) fulfilling a = f x( x, ŷ) + â. Since (g ( x, ŷ), h ( x, ŷ)) is surjective and Ω = Y holds we deduce from statement (i) of Corollary 2.1 the existence of (λ, µ) (K {o Z } + {(g( x, ŷ), h( x, ŷ))}) D, which equivalently means λ (K + {g( x, ŷ)}) D and µ Z, so that (â, f y( x, ŷ)) = (g ( x, ŷ), h ( x, ŷ)) [(λ, µ)] holds. Once again we make use of the definition of the adjoint operator to point out that for any x X and y Y the following is true: (x, y), (â, f y( x, ŷ)) = (x, y), (g ( x, ŷ), h ( x, ŷ)) [(λ, µ)] On the other hand we have: = (g ( x, ŷ), h ( x, ŷ))[x, y], (λ, µ) = g ( x, ŷ)[x, y], λ + h ( x, ŷ)[x, y], µ = g x( x, ŷ)[x], λ + g y( x, ŷ)[y], λ + h x( x, ŷ)[x], µ + h y( x, ŷ)[y], µ = (g x( x, ŷ) λ + h x( x, ŷ) µ)[x] + (g y( x, ŷ) λ + h y( x, ŷ) µ)[y]. (x, y), (â, f y( x, ŷ)) = x, â + y, f y( x, ŷ) = â[x] f y( x, ŷ)[y]. 11

16 Combining the last two equalities gives us â[x] = (g x( x, ŷ) λ + h x( x, ŷ) µ)[x] + (f y( x, ŷ) + g y( x, ŷ) λ + h y( x, ŷ) µ)[y] for any x X and y Y. We derive two conditions for the pair (λ, µ): f y( x, ŷ) + g y( x, ŷ) λ + h y( x, ŷ) µ = o Y, (18) â = g x( x, ŷ) λ + h x( x, ŷ) µ. (19) Since λ is an element of (K + {g( x, ŷ)}) D we deduce from the definition of the dual cone ξ + g( x, ŷ), λ 0 for arbitrarily chosen ξ K. Obviously, we have g( x, ŷ), λ 0 because o W belongs to K. Furthermore, we have g( x, ŷ) K and since K is a cone 2g( x, ŷ) K, particularly. That is why g( x, ŷ), λ = 2g( x, ŷ) + g( x, ŷ), λ 0 and especially g( x, ŷ), λ = 0 holds. We reason λ K D. Finally, we sum up (18), g( x, ŷ), λ = 0 and (λ, µ) K D Z to figure out (λ, µ) Λ( x, ŷ). Taking (19) into account we arrive at which gives us (λ, µ) Λ( x, ŷ): a = f x( x, ŷ) + g x( x, ŷ) λ + h x( x, ŷ) µ (20) c ϕ( x) cl ( conv ( {f x( x, ŷ) + g x( x, ŷ) λ + h x( x, ŷ) µ (λ, µ) Λ( x, ŷ)} )) since (17) holds. Now we apply Lemma 1.1 substituting E = X, F = W Z, p = f x( x, ŷ), q = (g x( x, ŷ), h x( x, ŷ)) as well as A = Λ( x, ŷ) to see that the convex hull can be dropped. If W is reflexive, the weak closure can be dropped as well since Z is finite-dimensional and hence a reflexive Banach space. That is why we finally arrive at formula (16) and the corresponding remark. (ii) We proceed similarly to the proof of (i). Firstly, we use statement (ii) of Corollary 2.2 and Lemma 3.1 in order to verify that ϕ is locally Lipschitz continuous at x. Furthermore, we apply Theorem 7 in [11] to show (17). For any a {f x( x, ŷ)} + DN Γ( x, ŷ)(f y( x, ŷ)) there is â DN Γ( x, ŷ)(f y( x, ŷ)) satisfying a = f x( x, ŷ) + â. Since PKRZCQ implies KRZCQ, we can apply statement (ii) of Corollary 2.1 in order to verify that there are λ (K + {g( x, ŷ)}) D, µ Z and q N Ω (ŷ) so that (â, f y( x, ŷ)) = (g ( x, ŷ), h ( x, ŷ)) [(λ, µ)] + (o X, q) holds. Similar calculations as used for the proof of (i) show that (19) and additionally f y( x, ŷ) + g y( x, ŷ) λ + h y( x, ŷ) µ = q (21) are satisfied. Let (d k ) cone(ω {ŷ}) be any sequence converging to d Y. Since N Ω (ŷ) = (Ω {ŷ}) D = (cone(ω {ŷ})) D holds we deduce from (21) for arbitrary k N: (f y( x, ŷ) + g y( x, ŷ) λ + h y( x, ŷ) µ)[d k ] = ( q)[d k ] 0. (22) Obviously, the functional q is continuous which means that d satisfies (22) as well. As a consequence, (9) holds with λ 0 = 1. Exactly in the same way as above we show λ K D and g( x, ŷ), λ = 0. That means again that (λ, µ) is a regular Lagrange-multiplier of (1) at ( x, ŷ), hence an element of Λ( x, ŷ) and finally we conclude (20). Since (17) holds, we show formula (16) again by applying Lemma 1.1. # 12

17 Note that the assumptions of Theorem 3.1 guarantee that the following approximation formula for the basic subdifferential of ϕ at x holds: B ϕ( x) {f x( x, ŷ) + g x( x, ŷ) λ + h x( x, ŷ) µ (λ, µ) Λ( x, ŷ)}. Finally, we want to illustrate the presented theorem by means of the following example. Example 3.1 Let X = R, Y = L 2 2 [0, 1], W = R R, K = R+ 0 R+ 0 and Ω = Y be fixed. We omit any equality constraints here and consider problem (1) given as follows: x R y = (y 1, y 2 ) L 2 2[0, 1]: f(x, y) = g(x, y) = (y 2 (t) + 1) 2 dt (y 2 1(t) y 2 (t), x y 2 (t))dt. The optimal value function ϕ of this problem is Lipschitz continuous around the point x = 0 and c ϕ( x) = [0, 2] holds. Futhermore, we can use Theorem 3.1 to acknowledge these results. Observe that for any x R we can specify a global optimal solution y(x) as follows: { (y 1 0, y 2 x) if x 0 x R: y(x) = (y 1 0, y 2 0) if x < 0. That is why we can calculate the optimal value function ϕ in the following way: { (x + 1) 2 if x 0 x R: ϕ(x) = 1 if x < 0. Obviously, ϕ is a non-differentiable but locally Lipschitz continuous function since it can be represented as the maximum of two continuously differentiable functions. Furthermore, we get { d R: ϕ ( x; d) = 2d if d 0 0 if d < 0 and as a consequence c ϕ( x) = [0, 2] from the definition of the Clarke subdifferential. In order to verify the observations above in terms of Theorem 3.1 we have to find a point ŷ Ψ( x) at first. For that purpose we use ŷ = y( x) = o L 2 2 [0,1]. Note that since dom(ψ) = R we have x int(dom(ψ)) automatically. For any sequence (x k ) R converging to x we define a sequence (y k ) via y k = y(x k ) for any k N. Since y k always is an element of Ψ(x k ) and (y k ) converges to ŷ we get that Ψ is inner semicontinuous at ( x, ŷ). It can be shown that f and g are continuously Fréchet differentiable at ( x, ŷ) with the Fréchet derivatives given as follows: r R d = (d 1, d 2 ) L 2 2[0, 1]: f x( x, ŷ)[r] = 0 f y( x, ŷ)[d 1, d 2 ] = g x( x, ŷ)[r] = (0, r) g y( x, ŷ)[d 1, d 2 ] = d 2 (t)dt ( d 2 (t), d 2 (t))dt.

18 Since we have g y( x, ŷ) [ L 2 2[0, 1] ] + cone(k + {g( x, ŷ)}) = L({(1, 1)}) + cone(k) = L({(1, 1)}) + K = W, PKRZCQ is satisfied at ( x, ŷ). Let λ = (λ 1, λ 2 ) K D = K solve the system (9), (10) which takes the form d = (d 1, d 2 ) L 2 2[0, 1]: 1 0 (2 λ 1 λ 2 )d 2 (t)dt = 0 0 λ λ 2 = 0 since PKRZCQ guarantees λ 0 R +. Then, obviously, λ conv({(2, 0), (0, 2)}) = Λ( x, ŷ) holds. Furthermore, we want to point out that the Banach spaces W, X and Y are reflexive spaces. Additionally, K is SNC everywhere since it is a subset of a finite-dimensional Banach space. As a consequence the conditions of Theorem 3.1 (statement (ii)) are fulfilled, which means that ϕ indeed has to be locally Lipschitz continuous around x and (16) must hold. We get from that formula c ϕ( x) {f x( x, ŷ) + g x( x, ŷ) λ λ Λ( x, ŷ)} = {0 + λ 2 (λ 1, λ 2 ) conv({(2, 0), (0, 2)})} = [0, 2] which is actually the subdifferential we wanted to approximate. Note that for any choice of the parameter x the feasible set Γ(x) of the given optimization problem is not bounded and hence a non-compact set. 14

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